Answer
The answer is below.
Work Step by Step
Given
\[
\tan \left(y^{2}+x^{2}\right)=w
\]
and then
\[
\sec ^{2}\left(y^{2}+x^{2}\right) \cdot 2 x=w_{x}
\]
and
\[
\begin{array}{c}
w_{x y}=2 \sec \left(x^{2}+y^{2}\right) \cdot \sec \left(x^{2}+y^{2}\right) \tan \left(x^{2}+y^{2}\right) \cdot 2 y \cdot 2 x \\
=8 x y \sec ^{2}\left(x^{2}+y^{2}\right) \tan \left(x^{2}+y^{2}\right)
\end{array}
\]
On the other hand,
\[
\sec ^{2}\left(x^{2}+y^{2}\right) \cdot 2 y=w_{y}
\]
,
\[
\begin{array}{c}
2 \sec \left(x^{2}+y^{2}\right) \cdot \sec \left(x^{2}+y^{2}\right) \tan \left(x^{2}+y^{2}\right) \cdot 2 x \cdot 2 y =w_{y x}\\
=8 x y \sec ^{2}\left(x^{2}+y^{2}\right) \tan \left(x^{2}+y^{2}\right)
\end{array}
\]
Thus, the given statement is true.
